Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 55 / 1

Let $\delta =(\rho -\bar{\rho })/\bar{\rho }$ and define the cosmological density power spectrum by $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }_{\mathrm{D}}(\mathbf{k}-{\mathbf{k}}^{\prime })P(k)$. The contribution per logarithmic wave-number interval is the dimensionless cosmological power spectrum ${\Delta }^2(k)=k^{3}P(k)/(2{\pi }^2)$. The linear cosmological mass variance on mass scale $M$ iswith $R$ a comoving smoothing radius and $W$ the top-hat filter in Fourier space. On halo scales the observed spectrum is consistent with greater variance at smaller masses. Growth therefore brings smaller objects to the collapse threshold earlier, statistically; larger structures assemble by accretion and dark-matter halo mergers. This statistical growth from smaller bound systems to larger ones is hierarchical galaxy formation. It is an ordering of dark-matter assembly, not a rule that every small visible galaxy must precede every large visible galaxy.

For linear modes the linear growth factor gives $P(k,z)=D^2(z)P(k,0)$. During matter domination $D\propto a$, so $P\propto a^2$ and the characteristic nonlinear mass increases. Once modes become nonlinear, coupling between scales and halo formation change the shape, so the same multiplication cannot be used for the entire late-time spectrum. At low redshift accelerated expansion suppresses linear growth. Galaxy clustering traces the matter spectrum with galaxy bias; it should not be equated directly with an unbiased matter measurement.

The broad linear shape was set by the cosmological transfer function before and around matter-radiation equality, with baryonic acoustic structure also imprinted before recombination. A nearly scale-invariant primordial curvature spectrum has ${\mathcal{P}}_{\mathcal{R}}(k)\propto k^{n_s-1}$, with $n_s$ close to one. After converting curvature perturbations to matter-density perturbations,Thus nearly scale-invariant primordial curvature does not mean a constant density $P(k)$. Modes with $k\ll k_{\mathrm{eq}}$ enter the horizon after equality and have $T\simeq 1$. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination, when radiation controls the expansion and cold-matter perturbations grow only slowly. The cold-dark-matter transfer function behaves approximately as $T\propto \ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ at large $k$. Hence the density spectrum turns over near $k_{\mathrm{eq}}=a_{\mathrm{eq}}{H}_{\mathrm{eq}}/c$:The turnover records the equality horizon, while the late nonlinear excess records gravitational clustering. On galactic scales the effective slope is greater than $-3$, giving the growing small-scale variance needed for hierarchical galaxy formation.

Cold dark matter has negligible primordial thermal velocities and a very short collisionless free streaming length on galactic scales. Warm dark matter has appreciable residual velocities while structure is being seeded; particles stream across small fluctuations and reduce their contrast. Its cosmological transfer function is consequently cut off below a characteristic length, suppressing low-mass halos and delaying their formation. Above that cutoff its assembly can still be hierarchical. The important distinction is the free-streaming scale, rather than the present temperature or an arbitrary particle-mass label.

Linear evolution assumes $|\delta |\ll 1$. When $\delta$ becomes of order unity, overdense regions depart strongly from the Hubble flow and can turn around and collapse. Collisionless dark matter develops multistream motion after trajectories cross; gravitational mixing redistributes energy and produces a bound dark-matter halo. The formal infinite-density collapse of an ideal spherical pressureless solution is not the physical endpoint. A roughly virialized halo has $2K+W\simeq 0$ and a characteristic virial velocity $V_{\mathrm{vir}}^2=G{M}_h/r_{\mathrm{vir}}$. Its gas virial temperature is conventionallyIt measures the thermal energy associated with the gravitational potential; the numerical factor depends on the velocity-dispersion convention. It does not imply that the collisionless dark matter has a thermodynamic gas temperature.

The unheaded request about baryon conversion efficiency of a halo is also answered here. Define $f_{\ast }=M_{\mathrm{stars}}/(f_b{M}_h)$ using the cosmic baryon fraction $f_b$. In small halos, shallow potentials let stellar feedback drive outflows or repeatedly heat star-forming gas; supernova energy per stellar mass is roughly fixed while binding energy per gas mass scales as $V_{\mathrm{vir}}^2$. Photoheating during reionization also prevents very small halos from retaining or accreting cool gas. Molecular/atomic cooling thresholds further reduce star formation in the smallest systems. These effects make $f_{\ast }$ fall toward low mass.

Near $M_h\sim 10^{12}{M}_{\odot }$, gas can cool efficiently and the potential is deep enough to retain more of it, while a long-lived hot atmosphere and maintenance heating are less effective than in larger systems. At high mass, higher virial temperature and lower cooling efficiency let a substantial hot atmosphere persist. Active-galactic-nucleus feedback can prevent that atmosphere from supplying cold gas and can expel some gas; the cooling-time bottleneck alone is not an adequate explanation for the low stellar fractions of massive groups and clusters. The peak reflects a competition between gas supply/cooling and feedback, rather than complete conversion of all baryons at a sharply universal mass. Its exact location and height depend on epoch, metallicity, gas history and the stellar population included.

If gas cannot radiate enough energy to fall below the virial temperature, infall converts gravitational energy into heat through shocks and compression. Thermal pressure can support an extended atmosphere in the dark-matter halo. It need not have the same density profile as the collisionless dark matter, because its entropy and pressure matter.

A useful quantitative comparison is the radiative gas cooling time $t_{\mathrm{cool}}\sim (3/2)n{k}_{B}T/(n_e{n}_i\Lambda )$ against the halo dynamical time $t_{\mathrm{dyn}}\sim r_{\mathrm{vir}}/V_{\mathrm{vir}}$. If cooling remains slow or a heating source balances it, the gas stays predominantly hot and diffuse, rather than forming a compact, cold, self-gravitating stellar system. Without sufficient cooling, most baryons remain pressure-supported halo gas. Very slow cooling can still feed gradual central condensation; the condition is about energy loss relative to the evolution time, not an absolute prohibition on any inward motion.

If the gas can cool appreciably below the virial temperature, radiative cooling removes thermal energy and pressure support. In a dark-matter halo the gas then contracts, dissipating more energy as it falls. Efficient condensation requires the radiative gas cooling time to be short enough compared with the relevant dynamical or assembly time; merely having an available low-temperature transition does not guarantee that the gas reaches it quickly.

The collisionless dark matter cannot lose comparable energy through radiation and remains extended. Gas with appreciable conserved angular momentum stops radial collapse when rotation supports it, often forming a disk; lower-angular-momentum gas reaches a more compact central region. Cold dense gas can fragment into self-gravitating clouds and form stars if its gravitational instability overcomes remaining support. Stellar feedback subsequently reheats or expels gas and regulates the conversion. Efficient cooling enables central baryonic condensation and star formation; angular momentum and feedback determine the resulting galaxy.

Below about $10^4\mathrm{K}$, neutral hydrogen electronic excitation becomes inefficient because the lowest relevant excitation energies greatly exceed the typical particle thermal energy. Primordial gas therefore needs molecular hydrogen cooling through rotational and vibrational transitions; HD can cool still colder gas where it is sufficiently abundant. Without molecules or metals, cooling can stall near the atomic threshold.

In enriched gas, metal-line cooling from low-energy fine-structure transitions, notably singly ionized carbon and neutral oxygen, remains effective below that threshold. At higher densities, molecular rotational lines such as CO and energy transfer from gas to dust followed by dust infrared emission are important. Cold-gas cooling is primarily molecular, fine-structure, or dust-mediated, according to composition and density. Molecule formation, dissociating radiation and the Cosmic microwave background temperature floor constrain how far cooling proceeds.

In the intermediate-temperature interval, atomic line cooling is generally efficient. Collisional excitation of hydrogen and helium followed by photon emission removes thermal energy; collisional ionization and subsequent recombination also contribute. As the gas becomes more ionized, different transitions enter and leave the cooling budget.

For enriched gas, metal-line cooling is often dominant over substantial parts of this interval, because heavy ions provide many ultraviolet and optical transitions. The cooling curve consequently has pronounced peaks rather than one smooth universal power law. Hydrogen/helium atomic processes and metal lines provide the main cooling channels here. Their relative strengths depend on metallicity, ionization state, density and the incident radiation field; these temperature bands describe typical gas, not composition-independent boundaries.

In sufficiently hot ionized gas, electrons radiate when accelerated in ion Coulomb fields: thermal bremsstrahlung is the main continuum cooling process. In the optically thin nonrelativistic limit its emission rate scales approximately asFor hot metal-poor gas this is the principal high-temperature channel. Metal ions still give important metal-line cooling near $10^6\mathrm{K}$, and in enriched gas can remain important up to several million kelvin; one should not infer that crossing $10^6\mathrm{K}$ instantly eliminates all lines. At sufficiently high temperatures most ions are stripped and free-free emission dominates. Inverse Compton cooling can also matter for ionized, diffuse gas in a strong radiation field, especially the high-redshift Cosmic microwave background. The high-temperature asymptote is bremsstrahlung cooling, with metal-line and Compton qualifications where appropriate.